Studdy AI Logo
Math  /  Trigonometry

QuestionGiven cos⁡θ=338\cos \theta=\frac{3 \sqrt{3}}{8}, what is sin⁡θ\sin \theta ? sin⁡θ=[?]\sin \theta=\frac{\sqrt{[?]}}{}

Studdy Solution

STEP 1

1. We are given cos⁡θ=338\cos \theta = \frac{3 \sqrt{3}}{8}.
2. We need to find sin⁡θ\sin \theta using the Pythagorean identity.
3. The Pythagorean identity states that sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.

STEP 2

1. Use the Pythagorean identity to find sin⁡2θ\sin^2 \theta.
2. Solve for sin⁡θ\sin \theta.

STEP 3

Start with the Pythagorean identity:
sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
Substitute the given value of cos⁡θ\cos \theta into the equation:
sin⁡2θ+(338)2=1\sin^2 \theta + \left(\frac{3 \sqrt{3}}{8}\right)^2 = 1

STEP 4

Calculate (338)2\left(\frac{3 \sqrt{3}}{8}\right)^2:
(338)2=(33)282=9×364=2764\left(\frac{3 \sqrt{3}}{8}\right)^2 = \frac{(3 \sqrt{3})^2}{8^2} = \frac{9 \times 3}{64} = \frac{27}{64}
Substitute back into the equation:
sin⁡2θ+2764=1\sin^2 \theta + \frac{27}{64} = 1

STEP 5

Solve for sin⁡2θ\sin^2 \theta:
sin⁡2θ=1−2764\sin^2 \theta = 1 - \frac{27}{64}
Convert 1 to a fraction with a denominator of 64:
1=64641 = \frac{64}{64}
Subtract the fractions:
sin⁡2θ=6464−2764=3764\sin^2 \theta = \frac{64}{64} - \frac{27}{64} = \frac{37}{64}

STEP 6

Take the square root of both sides to find sin⁡θ\sin \theta:
sin⁡θ=±3764\sin \theta = \pm \sqrt{\frac{37}{64}}
Simplify the square root:
sin⁡θ=±378\sin \theta = \pm \frac{\sqrt{37}}{8}
The value of sin⁡θ\sin \theta is:
sin⁡θ=±378\sin \theta = \pm \frac{\sqrt{37}}{8}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord